Comparison of Experimental and Computational Simulations Results of a Pulsed Plasma Accelerator

IEPC-2005-008

th Presented at the 29 International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005

T. Moeller*, D. Keefer†, R. Rhodes‡, and D. Rooney§ University of Tennessee Space Institute, Tullahoma, TN, 37388, USA

and

** D. Li and C. Merkle†† Purdue University, West Lafayette, IN 47907, USA

High power pulsed plasma thrusters (PPT) contain regions of high density plasma in close contact with regions of vacuum that challenge accurate numerical simulations using current MHD codes. A new version of the GEMS code has been developed to provide a method to accurately calculate these regions of disparate density. The GEMS code now includes a full Maxwell equation solver capable of providing both wave and diffusion solutions, and comparisons with analytical solutions and MACH2 simulations are given to demonstrate its accuracy. In addition, numerical calculations using GEMS illustrate the importance of a solution technique that handles all regimes from ideal insulator to good conductor. A MACH2 simulation of a simple Marshall gun thruster demonstrates the need for multiple regime computations in high power pulsed plasma thrusters. A laboratory Marshall gun thruster was developed to provide an array of diagnostic measurements that can be compared with the numerical simulations. The prototype thruster has been tested and a collection of photographic images, laser interferometer data, Rogowski coil data, and B-dot probe data was acquired. These measurements will provide data to compare with MACH2 and GEMS simulations.

Nomenclature r B = magnetic induction β = pseudo-time scaling coefficient c = speed of light r E = electric field v J = current density

* Research Associate Professor, Mechanical Engineering, Center for Laser Applications, [email protected]. † Professor Emeritus, Engineering Science, Center for Laser Applications, [email protected]. ‡ Research Scientist, Center for Laser Applications, [email protected]. § Graduate Research Assistant, Aerospace Engineering, [email protected]. ** Research Associate Professor, Mechanical Engineering, [email protected]. †† Reilly Professor of Engineering, Aerospace Engineering, [email protected]. 1 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 L = characteristic length scale t = physical time 2 VNN = von Neumann number = ∆t / µ0σ∆x

ε0 = electric permittivity φB = Lagrange multiplier for divergence constraint on B field φE = Lagrange multiplier for divergence constraint on E field ρE = electric charge density σ = electrical conductivity τ = pseudo time µ0 = magnetic permeability Subscripts R = reference quantity Superscript n = physical time step k = pseudo-time step * =dimensionless variable

I. Introduction Envisioned interplanetary missions that involve on-board space nuclear power systems will provide unprecedented levels of electrical power for a variety of mission needs including planetary base support, surface exploration and operations, life support and communications. The availability of this high power capability, however, also represents a challenging opportunity for the development of new, high power electric propulsion systems whose high specific impulse and high thrust levels can shorten trip times and gross vehicle weight, thereby increasing the overall attractiveness of the mission. Highly developed conventional EP thrusters such as ion engines, Hall thrusters and arcjets are well suited for lower power applications but have not been scaled to the power levels anticipated with fission-based space nuclear electric power systems. These smaller thrusters utilize electrostatic and thermal processes to provide thrust from a plasma propellant. Electromagnetic acceleration of plasmas for propulsion has long been seen as a means of providing efficient high specific impulse propulsion systems, however, development of this class of accelerators has languished because they are efficient only at power levels that heretofore have been unavailable in space. Two distinct types of electromagnetic thrusters have been suggested for the high power arena: magnetoplasmadynamic (MPD) thrusters, which are steady flow devices, and Pulsed Plasma Accelerators (PPA) that are variants of the Marshall Gun. These devices operate in different magnetic interaction regimes. The MPD operates at relatively small (Rm) while the PPA operates at large Rm. Pulsed systems are particularly of interest because they can be operated over a wide range of available power by varying the pulse duty cycle to retain efficient high-power operation while controlling average power. In the present paper we present initial results from a companion experimental/computational effort aimed at addressing the physical behavior and performance characteristics of high power pulsed plasma thrusters for possible applications in nuclear electric propulsion systems. The experimental portion of the program utilizes a laboratory prototype PPA that is designed to provide both detailed experimental diagnostics of PPA operation and accurate diagnostic data for validation of the companion computational simulations. The computational effort involves detailed simulations with two codes: the widely accepted MACH2 plasma dynamics code which is based on the MHD approximation and a somewhat simplified fluid dynamics capability, and a new in-house Maxwell solver known as GEMS that incorporates complete fluid dynamics capabilities and allows either MHD or full Maxwell solutions. One important role of the MACH2 code is to serve as a standard against which simulations from the newer GEMS code can be verified and to provide a historical link to the wide array of MACH2 calculations which have previously been done in our laboratory and elsewhere. Both MACH2 and GEMS are used for data interpretation and guidance to the experiments with the intention being to transfer more and more of this role to GEMS as its capabilities and consistency with MACH2 are verified. In addition, GEMS is being used to understand physics issues in areas where the MHD approximation is invalid and MACH2 cannot be used directly. The organization of the paper is a follows. Section II discusses the PPA configuration of interest and presents the details of the experimental laboratory prototype and some initial representative experimental data. Preliminary simulations of PPA operation from the MACH2 code are also presented in this section. In section III we present the electromagnetic implementation that is being used in GEMS with emphasis on a hyperbolic solution technique that is applicable to either the MHD equations of the complete Maxwell set. Section IV then presents the results of some 2 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 simple representative test problems that have been obtained from GEMS along with validation against closed form analytical solutions and companion MACH2 calculations. Section V summarizes our findings.

II. Experiment To provide validation data for computer simulations, we are developing a laboratory prototype pulsed plasma accelerator with an emphasis on providing accurate diagnostics as opposed to optimized thruster performance. MACH2 simulations were utilized to guide the design of this prototype. This thruster has been successfully fired, and ongoing testing is providing data with which planned simulations can be compared. The experimental setup and representative data from early testing are presented in this section.

A. Experimental Setup The thruster consists of a cylindrical center body 2-inches in diameter inside of a coaxial 4-inch diameter tube (Fig. 1). This outer tube is divided into a 2-inch upstream section and a 5-inch downstream section by a quartz insulating ring. The ring has sputtered metal strips connecting the two sections of the tube which, when vaporized by an ignition power supply, provides the plasma propellant that triggers and then is driven by the main power supply. The Quartz ring 17.5 µF 40 kV capacitor main power source, a 17.5 µF 40 kV capacitor charged with a 30 kV power supply, provides a maximum energy storage of almost 8 kJ with a calculated discharge time of 10 µs and an average power of nearly 800 megawatts1. Early system firing resulted in a fuse plasma that was not azimuthally uniform. To promote azimuthal symmetry, we placed axial slots in the outer electrode between the quartz fuse ring and the downstream trigger capacitor connection (Fig. 2). These slots create an array of parallel inductances between the fuse and the trigger power supply that act as an inductive divider to drive the current flowing through the electrode towards uniformity. To further promote the Thruster electrodes generation of a uniform discharge, the inner electrode was modified to include 1/8” wide slots that run almost the entire Figure 1. Photograph of prototype pulsed length of the electrode. This plasma accelerator (before adding axial slots). creates an inductive divider through which the main capacitor current must pass immediately following current initiation, rather than being delayed until the current sheet reaches the slotted section of the outer electrode downstream of the quartz fuse ring. The thruster is installed in a vacuum chamber fabricated from stainless steel tubing and plate (Fig. 3). The plasma discharges through a section Figure 2. Photograph showing details of slotted outer (left) and and inner of vacuum chamber with flat (right) electrodes. quartz windows that are used for interferometric, photographic, and spectroscopic measurements. An Imacon high speed camera is positioned to take photographs at up to 20 million frames per second. A single beam, heterodyne laser interferometer has been installed to obtain line- of-sight electron number density data downstream of the exit plane. Rogowski coils have been installed to monitor the current in the trigger and main capacitor circuits. Thirteen B-dot probes (see the left photograph in Fig. 2) have been installed on the outer electrode and are being utilized to help characterize the development of the current sheet inside of the thruster. The B-dot probes are connected to, 20 MHz data acquisition channels.

3 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 Fiber Optic

H.V. H.V. Trigger High Power Power Capacitor Speed Supply Supply Camera

Data Single Beam H.V. Capacitor Acquisition Heterodyne System Laser Mirror Interferomter

Vacuum Fuse Disk Chamber Fiber Optic

Rogowski coil, B-dot probe data

Figure 3. Schematic of Experimental Setup.

B. Preliminary Thruster Data Example outputs from the diagnostics were collected in tests with the main high 20000 voltage capacitor charged to 20 kV. The 15000 trigger current and main capacitor current as a function of time for this case are 10000 shown in Figs. 4 and 5, respectively. A 5000 high-speed photograph of the end-on view of the discharge is shown in Figure 6. 0 Current, A The frames in this figure were taken at a -5000 rate of 1 MHz with time progressing from -10000 bottom-to-top and from left-to-right. The peak current in the main capacitor is -15000 approximately 160,000 A and that in the 0.0E+00 5.0E-06 1.0E-05 1.5E-05 2.0E-05 trigger circuit is approximately 16,000 A. Time, s Representative path-integrated line-of- sight electron number densities calculated Figure 4. Time resolved trigger circuit current measured from data taken from the heterodyne laser using a Rogowski coil. interferometer2 are shown in Fig 7. 150000 Representative integrated B-dot probe 100000 data are shown in Fig. 8. Tracking the peaks in the probe data as a function of 50000 time along with the known probe positions 0 can be utilized to estimate an average -50000 velocity of various peaks in the current

Current, A sheet as it propagates in the thruster. The -100000 first peak is the best choice because it is -150000 the strongest and easiest to identify. -200000 Although analysis of the B-dot data is just 0.0E+00 5.0E-06 1.0E-05 1.5E-05 2.0E-05 commencing, preliminary results indicate Time , s that the average velocity of the current sheets ranges from roughly 480 km/s for Figure 5. Time resolved current in the thruster circuit the first current peak to roughly 59 km/s measured using a Rogowski coil. for the forth current peak. The decrease in

4 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 current sheet velocity with cycle peak number is attributed to that fact that a larger percentage of propellant mass is ejected from the thruster with each current cycle. This trend is predicted by preliminary MACH2 simulations that indicate an initial, low-mass puff of propellant exits the thruster at approximately 100 km/s, but the majority of the propellant mass exits the thruster at much lower velocities. Detailed comparisons between simulations and experimental data are not available for this paper but will be given in future publications. Figure 6. High-speed photograph of discharge (end-on view).

C. Preliminary MACH2 Thruster 3.50E+21

Simulation Results 3.00E+21 MACH2 is a time dependent (MHD) computer 2.50E+21 code that can be run in the axisymmetric 2.00E+21 mode3. When modeling with MACH2, there exists a class of problems where a 1.50E+21 conductor, usually a plasma, bridges the 1.00E+21 walls downstream of the point where the circuit is attached and an insulating gas or 5.00E+20 ‘vacuum’ fills the intervening space. In 0.00E+00

the corresponding physical device the 1/m^2 Density, Number Electron of Sight Line propagates across the -5.00E+20 vacuum as a wave. MACH2 does not -1.00E+21 model electromagnetic propagation by -2.53E-06 2.47E-06 7.47E-06 1.25E-05 1.75E-05 2.25E-05 2.75E-05 3.25E-05 waves, and some other mechanism is Time, s necessary to allow the current to move from the point of circuit attachment. The Figure 7. Line-of-sight electron number density determined transport mechanisms for a magnetic field from the heterodyne laser interferometer. included in the MACH2 MHD code are convection, diffusion, and the Hall Effect. 0.08 All of these have been used to transport 0.06 magnetic field across the vacuum with 0.04 differing degrees of success. During the past several years we at 0.02 UTSI have been experimenting with and 0 modeling pulsed plasma thrusters and, -0.02 more recently, a pulsed plasma accelerator -0.04 Current, Arbitrary (Fig. 9). In the PPA, the plasma is formed Magnetic Field, A -0.06 from an exploding foil which, after a short delay, bridges the annular gap discharging -0.08 the main capacitor. When modeling this -0.1 device we have chosen to use diffusion to -0.12 propagate the magnetic field from the 0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 capacitor connections to the plasma Time, s armature. To implement this we employ an ideal fluid we call ‘ether’ named after Figure 8. Sample integrated data from a B-dot probe the fluid which, before the development of located between the main capacitor and the fuse disk. wave theory, was believed to occupy all

5 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 space and provided the medium to propagate light. Our ether has a low molecular weight to 4 in minimize its effect on the total mass of propellant, a large constant resistivity to promote rapid diffusion, suppressed ohmic heating to avoid temperature runaway, and a Annular high viscosity to damp any vortices that might region before fuse trigger form and trap magnetic fields. 6 in Thrusters and opening switches where wave propagation may occur usually operate 2 in on microsecond time scales rather than the nanosecond or picosecond scales where electromagnetic waves propagate. The question usually addressed is how does the plasma move under the influence of the Fuse plasma expansion magnetic and fluid mechanical pressures, after trigger 2 in rather than the detailed distribution of the magnetic field. A calculation was made with MACH2 using a sinusoidal current input with a one microsecond rise time across the annular gap between a 50 mm and a 100 mm Figure 9. Schematic of laboratory prototype pulsed plasma tube. The tubes were shorted 50 mm from the thruster. inlet. Figure 10 shows rBθ in A as a function of time at the short and at the current connection. The von Neumann number (VNN) for this run is approximately 105, at least 3 orders of magnitude too large to give an accurate solution to the diffusion equation. The time for an electromagnetic wave to travel 50mm in vacuum is 166 picoseconds, negligible on this time scale. So the short would feel the inlet condition almost instantaneously if light speed propagation were available in the model. The magnetic field at the short initially lags, but even with the errors from a much too large a VNN, after 300 nanoseconds there is negligible difference in the magnetic field at the short between diffusion and light speed transmission. A computation of the thruster impulse at 1 microsecond shows about 0.5% difference between the two assumptions. It would be possible to run the initial phase at a smaller VNN and then increase the time step when the field stalls at the plasma armature. The problem occurs when the magnetic field breaks through the plasma and large magnetic fields try to move through regions of very low density. At this point the VNN increases at various times and places in the flow unless the time step is reduced, possibly to a point where the run cannot be completed. It may be necessary to run with large VNN’s to complete thruster 400000 -3 and opening switch calculations in an ∆x=.5x10 m -9 acceptable amount of time. With GEMS we ∆t=10 sec now have a tool where such regions may be η =108 m2 /sec 300000 5 examined in detail to evaluate the effects of vnn = 4 x 10 cruder approximations. The question of solving the diffusion equation with an excessive VNN was also 200000current, A examined by calculating the propagation of a magnetic field from a step function input of 400,000 A into a short 50 mm from the inlet current conection y = 0 mm (Fig. 11). The VNN was controlled by setting 100000 short y = 50 mm the maximum time step. The speed of propagation is reduced at larger time steps and at a VNN of 1000 the errors in the 0 solution become significant in this problem. 0 2E-07 4E-07 6E-07 8E-07 1E-06 Since this method is at best a crude time, sec approximation of wave propagation, a Figure 10. Current delay across a 50 mm insulating gap. computational method is needed that can incorporate both wave and diffusion 6 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 propagation of magnetic fields. 400000 An example of one PPA simulation that utilizes Teflon® as a propellant is presented below to demonstrate the time-evolution of the magnetic field and propellant mass 300000 distributions in such a device. A calculation using MACH2 was used to simulate a thruster t = 2 nanoseconds with a Marshall gun4 configuration with an

current, A ∆x=.5mm annular gap of 25mm confined by 3mm thick 200000 walls. In this simulation, the initial condition η =106 m2/sec employed an ionized Teflon® plasma (with a ∆ tnanosec vnn 3 density of 0.056 kg/m ) filling the annulus .1 1350 between 50 and 55mm axially. The gun is 100000 .01 135 driven by a 400,000 A sinusoidal current pulse .001 13.5 with a 1.3 µsec rise time. The circuit is .0004 5.4 connected between the coaxial electrodes at the upstream end of the thruster (bottom of Fig 0 0 0.01 0.02 0.03 0.04 0.05 12). The magnetic field is shown as 2πrBθ/µo distance, m in Fig. 12, and the region of high Teflon density is shown by two contour lines Figure 11. Effect of ∆t on solution accuracy. corresponding to 0.07 kg/m3 and 0.0005 kg/m3. At 1 µsec after current initiation the plasma has been slightly compressed and distorted from its original position, and the magnetic field is essentially confined by the plasma. At 1.25 µsec there is significant distortion of the plasma near the center-body where the Lorentz force is largest, since both the magnetic inductance, B, and the current density, J, decrease with radius. The magnetic field is still essentially confined by the plasma. At 1.5 µsec the magnetic field has broken through the plasma augmented by radial components of the Lorentz force that push the plasma toward the walls. At this point the magnetic field is again propagating into a vacuum and wave-like propagation must be included or simulated by another mechanism. Accurate modeling of these regions of widely varying plasma density will provide a tool which will allow a better understanding of the physical processes in a PPA. The electro-magnetic version of the GEMS code is being developed for this purpose.

1 µsec 1.25 µsec 1.5 µsec

2πrBrbthetaθ/µo, A 400000 375000 350000 0.1 0.1 0.1 325000 300000 275000 250000 m m m , , , 225000 200000 175000 y osition y y osition osition 150000 p p p 125000 100000 0.05 0.05 0.05

Axial 75000 Axial Axial 50000 25000 0

0.02 0.04 0.06 0.02 0.04 0.06 0.02 0.04 0.06 x x x Radius, m Radius, m Radius, m

Figure 12. Magnetic field and Plasma in a coaxial PPA.

7 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 III. Formulation of the Combined Parabolic/Hyperbolic Problem Maxwell’s equations for a linear, isotropic medium are, r 1 ∂E r r − ∇× B = −µ0 J (1) c 2 ∂t

r ∂B r + ∇ × E = 0 (2) ∂t

r ∇ • B = 0 (3)

r ∇ • E = ρe / ε0 (4) r r v where E is the electric field; B is the magnetic induction and J is the current density. The quantities, ε 0 and µ0 , are the electric permittivity and the magnetic permeability of free space, respectively; while c, is the speed of light; and ρe is the electric charge density. To express the current density in terms of the field quantities, we introduce a generalized Ohm’s law as r r r 1 r 1 r r E = −V × B + J + J × B (5) σ ne v where V is the plasma particle velocity, σ is the electrical conductivity, n is the electron number density, and e is the charge on an electron. Ohm’s law can be inverted to express the current density in terms of the electric and magnetic fields by defining a general electrical conductivity tensor, Λ , as, ⎛ σ σ ⎞ ⎜ 1 Bz − B y ⎟ ⎜ ne ne ⎟ 1 σ σ Λ = ⎜− B 1 B ⎟ (6) σ ⎜ ne z ne x ⎟ ⎜ σ σ ⎟ ⎜ B y − Bx 1 ⎟ ⎝ ne ne ⎠ to give, r r r r J = Λ−1 (E +V × B) (7)

Combining Eqs. 1 and 7 gives, r 1 ∂E r −1 r r r − ∇× B = −µ0Λ (E +V × B) (8) c2 ∂t

v v −1 The results in the present paper are based on the simpler form J = σE in which the tensor, Λ , reduces to the scalar, σ , and the velocity is taken as zero. Although this simpler form is used to demonstrate the approach, all conclusions can be interpreted in term of this more general formulation. There are three distinct temporal regimes in which solutions of Maxwell’s equations are of interest: (i) a quasi- v r steady regime where ∂E / ∂t = ∂B / ∂t = 0 and the partial differential equations reduce to the statements that both the r r divergence and curl of the vector B vanish while the curl of E vanishes, but its divergence depends on the space r r charge; (ii) a parabolic, MHD regime where (1/c2 ) ∂E / ∂t = 0, but ∂B /∂t ≠ 0 so that Maxwell’s equations reduce to v r the magnetic diffusion equation; and (iii) a hyperbolic, Maxwell regime where both ∂E / ∂t and ∂B / ∂t are non- zero and electromagnetic effects travel by wave-like processes. These three regimes can be illustrated by combining the above set of equations into a single equation for the magnetic field. Taking the curl of the first equation and combining with the temporal derivative of the second while using the divergence constraint on the magnetic field gives the convected wave equation: 8 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 2 r r 1 ∂ B ∂B 2 r + µ0σ = ∇ B (9) c2 ∂t2 ∂t The quasi-steady regime occurs when both the first and second temporal derivatives of B are negligibly small. The diffusion regime corresponds to the case where the source term, µ0σ , is large and the second-order temporal derivative can be omitted. Physically, this condition corresponds to applications in which the displacement current r r is negligible in comparison with the physical current, ε 0 ∂E / ∂t << J . When the source term is small, the displacement current must balance the curl of the magnetic field, and the equation takes the classical form of the wave equation. Succinctly, for small values of physical current where the displacement current is significant, disturbances travel as waves, but for large values of physical current and negligible displacement current, they propagate by diffusion as the above equation indicates. Although closed form solutions to these equations can be found for isolated cases in all three regimes, complex domains and general boundary configurations require numerical solutions. Our goal in the present paper is to present a single method that will allow solutions in all three regimes. Such a method is convenient for general applications, but is mandatory for propulsion applications such as those discussed in the Sections above where diffusion and wave-like regimes occur in different portions of the same problem. Before looking at the numerical solution of the general problem, we note that distinct numerical methods are widely used for both the diffusion and wave-like regimes. In the diffusion regime, which is generally associated with MHD problems, solutions are generally obtained by combining Faraday’s law and Ampere’s law to give the second-order magnetic diffusion equation5, 6. A key issue with such methods is identifying a technique for enforcing the divergence constraint because discrete forms of the div and curl operators do not generally satisfy the vector identity that div curl = 0. One method for enforcing this vector identity has been suggested by Powell7, and adapted by various others including ourselves8-10. Staggered grid methods represent an exception to this discrete inconsistency and can be used without a special divergence constraint. Wave equation solutions in the Maxwell regime are also widely used in both the time and frequency domains. Numerical methods in the time domain are generally based upon the first order equations as given above, and are restricted to problems in which the physical current is small.

A. Incorporating Pseudo-Time The magnetic diffusion equation constitutes an efficient description of problems for which the displacement current is negligible, but it fails when the displacement current becomes significant, since this term has been dropped altogether. Direct, time-marching solutions of the original first-order form of the equations appears attractive for problems in which both physical current and displacement current are significant, but these methods fail when the displacement current is small because the temporal derivative of the electric field disappears and leaves no update method for the E-field. Time-marching methods also encounter difficulties in enforcing the two divergence constraints which again have no temporal derivatives. To circumvent these difficulties, we use Lagrange multipliers to add two potential-like functions to the divergence constraints11 to bring the equation set to an eight-by- eight system and fashion an iterative method for this coupled system in terms of a pseudo-time which contains non- zero pseudo-time derivatives for all equations and therefore allows marching in pseudo-time. The pseudo-time corresponds to iterating to the proper transient condition at each physical time step. We denote the pseudo time by the symbol, τ , to distinguish it from the physical time, t. The modified Maxwell’s equations with the Lagrange multipliers, ϕB and ϕE , included and with pseudo-time derivatives added to all equations become: r r 1 ∂E ∂E r r + − ∇ × B + ∇ϕE = −µ0σE (10) βc2 ∂τ ∂t

r r ∂B ∂B r + + ∇ × E + ∇ϕ = 0 (11) ∂τ ∂t B

1 ∂ϕ r B + ∇ • B = 0 (12) βc2 ∂τ

9 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 ∂ϕ r 1 E + ∇ • E = ρ (13) ∂τ ε0 The pseudo-time enables us to use a common hyperbolic method to provide solutions in all three regimes. In the limit as the pseudo-time goes to infinity, the two potential-like functions go to zero and the divergence constraints are recovered for either steady or transient conditions. Efficient solution of the pseudo-time equations, however, requires the definition of a scaling coefficient, β , to ensure that the pseudo-time derivatives match the dominant terms in the equation in each of the three regimes. This choice can be made by non-dimensionalizing the terms in the equations in such a manner that they are balanced for each regime, and selecting β so that the pseudo-time derivatives are balanced with the physical terms.

B. Scaling the Pseudo-Time Equations We non-dimensionalize the equations by defining reference quantities denoted by subscript R, and dimensionless variables denoted by a superscript asterisk. The non-dimensional pseudo-time equations are:

r r B ∂B* B ∂B* E r ϕ R + R + R ∇ × E* + BR ∇*ϕ* = 0 (14) * * B τ R ∂τ tR ∂t L L

r r E ∂E* E 1 ∂E* B r ϕ r R + R − R ∇* × B* + ER ∇ϕ* = −µ σE E* (15) 2 * 2 * E 0 R βc τ R ∂τ tR c ∂t L L

ϕ ∂ϕ* B r BR B + R ∇* • B* = 0 (16) 2 * βc τ R ∂τ L

ϕ ∂ϕ* E r 1 ER E + R ∇* • E* = ρ (17) * τ R ∂τ L ε0

For the quasi-steady limit, we choose the reference B-field as BR ≡ µ0σLER so that the curl B term balances the physical current and the reference potential ϕER ≡ BR to balance the reference B-field. Ampere’s law then becomes, r 1 1 ∂E r r − ∇ × B + ∇ϕ = −E (18) 2 E µ0σc β τ R ∂τ where we have dropped the displacement current since we are interested in the quasi-steady condition. Applying this scaling to the remaining three equations and defining ϕBR ≡ ER gives: 2 r µ0σL ∂B r + ∇ × E + ∇ϕB = 0 (19) τ R ∂τ

1 1 ∂ϕ r B + ∇ • B = 0 (20) 2 µ0σc β τ R ∂τ

µ σL2 ∂ϕ r 1 L 0 E + ∇ • E = ρ (21) τ R ∂τ ε0 ER To ensure efficient convergence, the pseudo-time derivatives must balance the dominant physical terms. 2 −2 Accordingly, for the quasi-steady limit, we define, τ R = µ0σL and β = ()µ0σcL . This equation is properly 2 ordered so long as µ0σc tR >> 1. That is to say, it is pertinent for large characteristic physical times.

10 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 For the diffusion equation limit we balance the curl E term with the B-dot term in Faraday’s law by choosing the reference B-field as BR ≡ ERtR / L . Faraday’s law then becomes, r r tR ∂B ∂B r + + ∇ × E + ∇ϕB = 0 (22) τ R ∂τ ∂t

Using this reference condition in Ampere’s law then gives,

2 r 2 1 ⎛ L ⎞ tR ∂E r µ0σL r ⎜ ⎟ − ∇ × B + ∇ϕE = − E (23) β ⎝ ctR ⎠ τ R ∂τ tR where we have again dropped the displacement current. To order this equation properly we note that time-accurate solutions of a diffusion equation require that the 2 physical time step be chosen such that the von Neumann number, VNN = ∆t / µ0σ∆x be of order one. Accordingly, we choose the physical characteristic time as tR = ∆t and the characteristic length scale as the grid 2 size, L = ∆x , so that tR = ∆t = VNNµ0σ∆x . Balancing the remaining terms in the equation then gives, τ R = ∆t −2 and β = ()VNNµ0σc∆x . The scaling of the two potential-like functions is the same as in the quasi-steady case. The limit in which the displacement current is significant proceeds in like fashion. For this condition, the 2 reference B- and E-fields are related as, ER = BRtRc / L . Here we choose the physical characteristic time from the CFL value recognizing that the characteristic velocity is the speed of light when the full Maxwell solution is used. Therefore, we have, tR = ∆t = CFL∆x / c , where CFL is of order one. The scaling then gives τ R = L / CFLc and β = 1. The results of the scaling procedure are given in Table I for all three regimes. r r In employing the method, the full Maxwell equations (including the temporal derivatives of both B and E ) are retained at all points throughout the domain. The value of β is then defined for every cell within the computational domain according to local conditions. The hyperbolic pseudo-time iterative method is then used to converge the solution at each physical time step. Thus, for example, if the displacement current is significant over a portion of the computational domain it is evaluated throughout the domain, but has no impact in regions where it is small. For problems where the displacement is everywhere small, the user has a choice of omitting or including the displacement current in the entire domain. If it is truly negligible, the solutions will be essentially identical.

Table I Values of Scaling Factor, β Limit β 2 Quasi-Steady ⎛ 1 ⎞ ⎜ ⎟ Limit β = ⎜ ⎟ ⎝ µ0σcL ⎠ 2 MHD ⎛ 1 1 ⎞ ⎜ ⎟ Limit β = ⎜ ⎟ ⎝VNN µ0σc∆x ⎠ Maxwell β =1 Limit

C. Numerical Solution of the Equations The introduction of the pseudo-time derivatives allows us to iterate at each physical time step to ensure that the proper solution at each time step is obtained after a series of iterations. Because the pseudo-time is formulated in a hyperbolic form with a complete independent variable vector, we can use a hyperbolic method to perform the iterations at each time step even if the equations are parabolic in physical time. Although time-accurate calculations traditionally require a method that is second-order accurate, we here symbolically use first-order Euler implicit differencing in physical time to simplify the notation. For the physical time, we use a superscript n to indicate the

11 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 previous (known) time level and a tilde to denote the new (unknown) time level such that a representative time derivative becomes: ~ ~ ∂E E − E n = (24) ∂t ∆t After differencing in physical time, the equations become:

~ ~ n 1 ∂E 1 E − E ~ ~ t + − ∇× B + ∇ϕ E = −µ0σE (25) βc 2 ∂τ c 2 ∆t

r ~ ∂B B − B n ~ + + ∇× E + ∇ϕ~ = 0 (26) ∂τ ∆t B

1 ∂ϕ~ ~ B + ∇ • B = 0 (27) βc2 ∂τ

~ ∂ϕE ~ 1 + ∇ • E = ρe (28) ∂τ ε0 These equations are now discretized in pseudo time using the superscripts k and k+1 to give ~k +1 ~k ~k +1 ~k ~k n E − E E − E + E − E ~k +1 ~k +1 t k +1 + − ∇ × B + ∇ϕ = −µ0σE (29) βc2∆τ c2∆t E

r r ~ ~ ~ Bk +1 − Bk Bk +1 − Bk + Bk − Bn ~ + + ∇ × E k +1 + ∇ϕ~k +1 = 0 (30) ∆τ ∆t B

~ ~ ϕ k +1 −ϕ k ~ B B + ∇ • Bk +1 = 0 (31) βc2∆τ

ϕ~k +1 −ϕ~k E E ~k +1 1 k +1 + ∇ • E = ρe (32) ∆τ ε0 The spatial derivatives are discretized by a second-order accurate upwind, finite-volume process, and solved by an approximate Riemann solver12, 13. Representative results from this method are given in the following section.

IV. Simulation Results To demonstrate the capability of the GEMS code for the full Maxwell equations, we discuss numerical experiments where the Maxwell solver is applied in different situations. First, the diffusion of a magnetic field through a finite conductor in response to a step change in the magnetic field at the boundary is presented and compared with a corresponding analytical solution to demonstrate that the hyperbolic implementation produces accurate results for the diffusion problem. Following, we contrast the approximate diffusion solution to a complete Maxwell solution for cases where the diffusivity is small enough that wave processes becomes important. We then look at the magnetic field propagation through a domain composed of a vacuum region enclosed by finite conducting walls. As an approximation we fill the vacuum region with ‘ether’ having a finite conductivity and compare numerical solutions for the rate of diffusion of magnetic field through the ether as obtained with the MACH2 MHD code, which solves the magnetic diffusion equation, and the present GEMS formulation which can solve either the full Maxwell equations or the diffusion approximation. The results show excellent agreement between the present code and the MACH code, while also showing the dramatic differences in the manner in which magnetic fields propagate by wave processes sustained by displacement current, and in diffusion processes sustained by physical current.

12 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005

A. One-Dimensional Diffusion Problem As an initial verification of the computational method, we consider the temporal evolution of the electric and magnetic fields in a one-dimensional, semi-infinite domain, 0 ≤ x < ∞ . To enable comparison with an analytical r solution, the displacement current term, (1/ c2 )∂E / ∂t , is neglected so that the magnetic diffusion equation applies. r r The corresponding electric field is then given by the curl of the magnetic field, E = (∇ × B)/ µ0σ . As the initial condition, the electric and magnetic fields are set to zero throughout the domain. At time zero, the boundary condition on the axial component of the magnetic field at x = 0, steps from Bz = 0 to Bz = B0 and remains constant for all time. The other two magnetic field components at x = 0, and all three components as x → ∞ remain unchanged. Under these conditions the only non-zero components of the solution are the z-axial magnetic field, Bz , and the transverse electric field, Ey , and the magnetic diffusion equation reduces to a one-dimensional equation that possesses the similarity solution:

⎪⎧ ⎛ x µ σ ⎞⎪⎫ B = B 1− erf ⎜ 0 ⎟ B = B = 0 (33) z 0⎨ ⎜ ⎟⎬ x y ⎩⎪ ⎝ 2 t ⎠⎭⎪

1 ∂B 1 ⎛ x µ σ ⎞ E = x = exp⎜ 0 ⎟ E = Ez = 0 (34) y ⎜ ⎟ x µ0σ ∂x πµ0σt ⎝ 2 t ⎠ To provide a numerical solution for comparison with this analytical result we consider a two-dimensional solution of the first-order Maxwell equations with the displacement current omitted, r 1 ∂E r r − ∇× B = −µ0σE (35) βc 2 ∂τ

r r ∂B ∂B r + + ∇ × E = 0 (36) ∂τ ∂t The divergence constraints are automatically satisfied in the one-dimensional limit so the Lagrange multipliers are omitted. Numerical solutions for this first-order analog of the diffusion equation are obtained on a two-dimensional, uniform grid of size 176x105 for the finite domain 0 ≤ x ≤ 1, 0 ≤ y ≤ 0.3 with boundary conditions Bz = B0 , r r ∂Ey / ∂x = 0 on x= 0, along with a zero slope condition, ∂B / ∂n = ∂E / ∂n = 0 , on the remaining three sides. Applying a zero-slope condition at a finite location, x = 1, is slightly inconsistent with the analytical solution for which the magnetic field was set to zero as x → ∞ but is sufficient to allow verification of the numerical results. r r The initial condition is, B = E = 0 . Two cases are selected for verification against the analytical solution, σ = 10−2 mho/m and σ = 10−4 mho/m, 7 2 9 2 or η = 1/()µ0σ = 8×10 m /s and η = 8×10 m /s. The time step is selected by setting the von Neumann number, 2 −12 −14 vnn = ∆t /(µ0σ∆x ) = 2.4, yielding time step sizes of ∆t = 10 s and ∆t = 10 s, respectively. Comparisons with the 1-D analytical solution are presented in Figs. 13 and 14 where we show the field variation along the centerline of the 2-D domain. The results show very good agreement between the numerical and analytical solutions apart from the anticipated error near the downstream boundary (x = 1), where the two boundary conditions are dissimilar. The solutions also show that the wave diffusion speed is proportional to the electrical diffusivity, η .

13 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 −2 Figure 13. Comparison of GEMS and analytical solution for σ = 10 mho/m (Left: Bz ,

Right: E y ).

−4 Figure 14. Comparison of GEMS and analytical solution for σ = 10 mho/m (Left: Bz , Right: Ey ).

B. Magnetic Diffusion with Finite Displacement Current r We next repeat the σ = 10−2 mho/m calculation with the displacement current term, (1/ c2 )∂E / ∂t , included. When the displacement current is included, Maxwell’s equations for this problem simplify to the telegraph equation whose solution is composed of a precursor lossless wave followed by a residual diffusion wave that is left behind and decays exponentially with time after the initial pulse passes through. Similar characteristics are seen for the σ = 10−2 mho/m calculation in Fig. 15 when displacement current is included. The initial portion of the B-field that propagates through the region is composed of a wave front which is immediately followed by a diffusion process. Consequently, the nature of the transient B-field is completely different when displacement current effects are included. The diffusion solution from Fig. 14 is shown in blue for reference. The numerical experiments indicate that for the present problem, the displacement current affects the electromagnetic wave when the electrical conductivity of the media falls below approximately 1 mho/m. Under such conditions, the MHD assumption of a weakly ionized plasma is not accurate and the full Maxwell equation solver is necessary.

14 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 Figure 15. Comparison of numerical field solution with displacement current included (red) E with diffusion equation solution (grey) for σ = 10−2 mho/m (Left: Bz , Right: y ).

C. Electromagnetic field in two mediums The next case considers the transient electromagnetic field in an annular vacuum chamber that represents a simplified version of the PPT configuration discussed above. The length, L, of the vacuum region is 50 mm while the inner and outer walls are formed by concentric, stainless steel cylinders of radius r1= 22 mm and r2 = 53 mm with a wall thickness δ = 3 mm as shown in Fig. 16a. In place of the slug of plasma that serves as the propellant, the top of this annular region is enclosed by a stainless steel armature that carries the current from the inner to outer electrode. As noted above, the difficulty in this electromagnetic calculation is that the magnetic field must be carried by wave processes in the vacuum region, whereas in the stainless steel it is carried by diffusion. Hence, the solution of the complete problem requires capability for solving both the diffusion limit and the wave propagation limit simultaneously. As a first step in addressing this problem we begin by filling the vacuum region with an ‘ether’ medium of finite conductivity so that both the stainless steel and vacuum regions can be represented by the diffusion approximation. This allows us to again verify that the hyperbolic solution method employed in the GEMS code provides the same

Figure 16. Left: Geometry of concentric stainless steel channels with low conductivity material shown in blue replacing vacuum region in the center and by high conductivity material (stainless steel) shown in red. Right: Comparison of transient magnetic field propagation as computed by GEMS (red) and Mach (blue). 15 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 results as the solution of the more traditional magnetic diffusion equation that is obtained from MACH2. Because MACH2 is an MHD code that deals only with the magnetic diffusion equation, it cannot solve the present problem unless the vacuum region has a finite electrical conductivity to propagate magnetic field by diffusion. Following this diffusion calculation, we then include the displacement current effects in GEMS and recompute the solution with the conductivity in the vacuum set identically to zero to contrast the differences in the transient fields when they are propagated by displacement current rather than by physical current. For the ether solution we set the electrical conductivity in the vacuum region to σ = 10-2 mho/m, and specify the conductivity of the stainless steel walls as σ = 8x105 mho/m. The problem is then computed with both MACH2 and GEMS. The time step for the diffusion calculation is set at ∆t = 0.1 ps is used to catch the very fast initial wave diffusion phenomena in the vacuum (ether) region. Figure 16b compares the axial distribution of the current flow (the rBθ field) as a function of axial distance for six different times as predicted by GEMS and MACH2. The field is seen to propagate through the ether in exponential-like fashion. The very high conductivity in the stainless steel wall on the end of the domain as compared to the ether causes the slope of the curves to change discontinuously and to decay very rapidly to zero. On these very short time scales, the current diffuses to only a very thin skin depth in the stainless steel. The close agreement between the GEMS and MACH2 results is further evidence that the wave- like computational scheme in GEMS provides accurate solutions to problems involving pure diffusion. Contour plots of the rBθ field for this same case taken from the GEMS solution are presented in Fig. 17 for three time instants. These contour plots indicate that the current flow (the rBθ field) in the ether is essentially one dimensional with the only two-dimensional effects occurring in the skin effect region inside the stainless steel walls on the sides of the low conductivity region. The grids used in the stainless steel are not sufficiently refined to represent this skin effect accurately, but the plots show that the diffusion depth into the stainless steel is very small. At much longer times (results not shown) the entire current diffuses into the stainless steel and none of the current is carried by the low conductivity ‘ether’.

Figure 17. Transient current distribution in ‘ether’ region surrounded by stainless steel obtained with diffusion equation.

r Companion calculations of the same geometry with the displacement current term, (1/ c2 )∂E / ∂t , included and the conductivity in the vacuum region set to zero are presented in Figs. 18 and 19. Figure 18 compares the displacement current calculations (full Maxwell solutions) with the diffusion current solution (physical current only, no displacement current) previously shown in Fig. 16. As can be seen, there are significant differences between magnetic field propagation by diffusion and by wave processes. When the conductivity is zero, Maxwell’s equations reduce to a pure wave equation and disturbances are carried by a wave moving at the (constant) speed of light. By contrast, the speed of the diffusion wave is proportional to the electrical diffusivity and decreases as the inverse square root of time. Further, the amplitude of the diffusion wave is damped exponentially while the pure 16 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 wave propagating through the lossless medium maintains constant amplitude. For the conductivity value chosen here (σ = 10-2 mho/m), the diffusion process propagates considerably faster than speed of light and so reaches the end of the domain before the wave solution. (Setting the conductivity to 10-2 mho/m is arbitrary, and an ‘ether’ conductivity could be chosen to more closely match the speed of the wave solution, but no attempt to do so has been made here.)The qualitative character of the two families of rBθ field lines is also different. The field lines are concave downward for the wave solution, while they are concave upward for the diffusion solution, a further reflection that the two physical mechanisms of transport are based upon completely different physics. These numerical results demonstrate the Figure 18. Comparison of displacement current in vacuum importance of a solution technique that with simulated physical current in ‘ether’ medium. handles all regimes of operation. Representative contour plots of the rB Figure 18. Comparison of displacement current in vacuum θ with simulated physical current in ‘ether’ medium. field at three different times are given in Fig. 19 for the case in which the field is transported by the displacement current. These solutions are at the same times as those in Fig. 17 and give a direct comparison of the impact of the two methods of distributing magnetic field. It is important to note that the parameter, 2πrBθ / µ0 , represents the sum of both the physical current and the displacement current. In Fig. 17 the r displacement current is zero (the ∂E / ∂t term is omitted) while in Fig. 19 the physical current vanishes in the vacuum region (where the conductivity is zero). The wave solution shown in Fig. 19 does include current along that portion of the electrodes over which the magnetic wave has passed, but is zero in front of it. By contrast the surface current on the stainless steel in the diffusion solutions goes all the way around the closed circuit starting from time

Figure 19. Transient current distribution in vacuum region surrounded by stainless steel as a function of time. Solutions obtained with Maxwell equations including displacement current term.

17 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 zero, but with an exponentially small amplitude that increases with time until a final uniform current is established. The above diffusion and Maxwell solutions show major differences on a picosecond time scale, but the many of the key physics in PPA simulations occur on a microsecond time scale. It is important to ascertain whether any of these early time differences remain at later times or whether additional displacement current effects are introduced during the PPA cycle. Specific answers to these questions require detailed simulations of the entire pulse phenomena and careful comparisons with the companion experiments, but to give a first indication of differences at longer times, we here present some longer time simulations of the displacement current case for the present simplified problem. Clearly, at infinite time, both the wave solution and the diffusion solution will approach the same steady state condition with uniform magnetic field and zero current in the vacuum region and a linear magnetic field and uniform current density in the stainless steel. (The steady state diffusion solution will have a minute current in the vacuum region determined by the ratios of the conductivities in the stainless steel and ether.) The results in Figs. 16 and 17 can easily be extrapolated to steady conditions, whereas the wave solutions in Figs. 18 and 19 are so early in time that the approach to steady state cannot be anticipated. To address this issue, we present some longer time solutions in Fig. 20 and 21. Figure 20 shows 2πrBθ / µ0 profiles at various times in a fashion identical to that used in Fig. 18, but for time increments of 100 ps, starting from 100 ps and going to 1 ns. Looking first at Profiles 1 and 2, the shapes are similar to those previously shown in Fig. 18. When the wave reaches the higher conductivity material it reflects with approximately double the amplitude and then propagates back toward the boundary where the magnetic field was originally imposed (Profiles 3 and 4) where it again reflects and repeats the process (Profiles 5 through 10). Each succeeding reflection from the stainless steel transmits a small fraction of the energy into the stainless steel where diffusive effects dominate. This transmission of the magnetic field into the stainless steel results in a gradual amplitude decay in the vacuum region. The diffusion into the higher conductivity material can also be seen on the right side of the figure. This diffusion rate has Figure 20. Axial profiles of 2πrBθ / µ0 along centerline been amplified by decreasing the conductivity of vacuum and regions for various times from 100 to in the stainless steel to 800 mho/m for this 1000 ps. Stainless steel conductivity changed to 800 calculation. mhos/m. Figure 21 shows the time history of the magnetic field intensity ( 2πrBθ / µ0 ) at Point A in the inset as a function of time for the diffusion solution (Fig. 16) and the wave equation solution (Figs. 18 and 20). For the diffusion calculation the vacuum region has been filled with ether (σ =10−2 mho/ m ) and the displacement current is set to zero. For the wave solution, the conductivity in the vacuum is zero (σ =0) and displacement current is the only mechanism for propagating the magnetic field through the vacuum. For the case without displacement current, the magnetic field at Point A is seen to increase monotonically from zero to the imposed maximum in approximately 100 ps. The rate of increase is, of course, determined by the conductivity level in the ether. The wave solution, however, behaves much differently and approaches the steady solution through a damped oscillatory trajectory. As noted above, the damping rate is determined by the conductivity of the adjacent material. The present solution, which is based upon the same 800 mhos/m conductivity used for Fig. 20, approaches steady state conditions at about 20 ns. An analogous calculation with the conductor set at 8 mhos/m showed much more rapid damping and reached steady conditions at approximately 0.3 ns, whereas computational attempts at σ = 8×105mho/ m decayed exceedingly slowly (in line with an expected decay in 3 µs). The plasma armature in the PPA would be expected to have conductivities much lower than stainless steel, and the imposed magnetic field would vary with time rather than be constant, but the present results provide some understanding of the manner in which wave-transported fields differ from diffusion-transported fields. 18 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 Figure 21. Time history of Maxwell equation solutions with and without displacement current. Results correspond to magnetic field at Point A in inset.

V. Summary and Conclusions A laboratory prototype pulsed plasma accelerator that has been designed for accurate diagnostics rather than thruster performance to provide data with which our computer simulations can be compared has been described. The thruster has been successfully fired and the collection of photographic images, laser interferometer data, Rogowski coil data, and B-dot probe data is ongoing. These measurements will provide data with which MACH2 and GEMS simulations can be compared. Preliminary MACH2 simulations of the thruster have demonstrated the need for a full Maxwell equation solver such as the GEMS code that can accurately model wave propagation as well as propagation by diffusion in devices that have regions of dense plasma next to regions of near vacuum. To demonstrate the capability of the GEMS code for the full Maxwell equations, we utilized numerical experiments where the Maxwell solver is applied in three different situations. The first case included the diffusion of a magnetic field through a finite conductor in response to a step change in the magnetic field at the boundary. Comparisons between the GEMS and analytical solutions indicate very good agreement and also show that the wave diffusion speed is proportional to the electrical diffusivity. The second case involved comparisons of the approximate diffusion solution to a complete Maxwell solution for cases where the diffusivity is small enough that wave processes becomes important. Results of this case indicate that the displacement current affects the electromagnetic wave when the electrical conductivity of the medium falls below approximately 1 mho/m. Under such conditions, the MHD assumption of a weakly ionized plasma is not accurate and the full Maxwell equation solver is necessary. The final test case was one in which the magnetic field is propagated through a domain composed of a vacuum region enclosed by finite conducting walls. For this last case, GEMS solutions were compared with MACH2 solutions, and the results show excellent agreement between the present code and the MACH2 code, while also showing the dramatic differences in the manner in which magnetic fields propagate by wave processes sustained by displacement current, and in diffusion processes sustained by physical current. Future comparisons of experimental measurements from the laboratory prototype pulsed plasma accelerator and simulations from both GEMS and MACH2 are planned for the near future.

Acknowledgments This work was partially supported by the Arnold Engineering and Development Center under contract #F40600- 00-D-001 0024. The authors would like to thank the Propulsion Research Center at Marshall Spaceflight Center for 19 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005 the loan of the high voltage capacitors used in the experimental phase of this work.

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20 The 29th International Electric Propulsion Conference, Princeton University, October 31 - November 4, 2005